From the documentation on Standardize:

Based on that, and on the error message you received, it seems pretty clear that the problem is that you have columns in new whose values are equal, and so StandardDeviation returns 0. So, maybe Standardize is not doing what you want for 2D data, in which case, you'll need to transform your data somehow. Or maybe your data is malformed, so you'll need to figure out what's wrong with it.

Anyway, it doesn't seem like the problem is with the shape of your data but with the values of the data itself. But then, again, I don't really know the semantics of your situation, so I may be spouting nonsense.

Thank you for your answer. But the result needs 10 mean and 10 standard devieation. You can image that every column is a feature and every feature has it own mean and deviation.

ClearAll[standardizeTo]
standardizeTo[oldData_, newData_] :=
 Module[
   {oldMean, oldStdev},
   {oldMean, oldStdev} = Through[{Mean, StandardDeviation}[oldData]];
   oldStdev Standardize[newData] + oldMean
 ]

old = {{5,1,1,1,2,1,3,1,1,2},{5,4,4,5,7,10,3,2,1,2},{3,1,1,1,2,2,3,1,1,2},{6,8,8,1,3,4,3,7,1,2},
{4,1,1,3,2,1,3,1,1,2},{8,10,10,8,7,10,9,7,1,4},{1,1,1,1,2,10,3,1,1,2},{2,1,2,1,2,1,3,1,1,2},{2,1,1,1,2,1,1,1,5,2},
{4,2,1,1,2,1,2,1,1,2},{1,1,1,1,1,1,3,1,1,2},{2,1,1,1,2,1,2,1,1,2}};
new = {{4,8,6,4,3,4,10,6,1,4},{4,8,8,5,4,5,10,4,1,4}};

restandardized = standardizeTo[old, new];